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Mirrors > Home > ILE Home > Th. List > addneintr2d | GIF version |
Description: Introducing a term on the right-hand side of a sum in a negated equality. Contrapositive of addcan2ad 7942. Consequence of addcan2d 7940. (Contributed by David Moews, 28-Feb-2017.) |
Ref | Expression |
---|---|
addcand.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
addcand.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
addcand.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
addneintr2d.4 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
addneintr2d | ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addneintr2d.4 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
2 | addcand.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
3 | addcand.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
4 | addcand.3 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
5 | 2, 3, 4 | addcan2d 7940 | . . 3 ⊢ (𝜑 → ((𝐴 + 𝐶) = (𝐵 + 𝐶) ↔ 𝐴 = 𝐵)) |
6 | 5 | necon3bid 2347 | . 2 ⊢ (𝜑 → ((𝐴 + 𝐶) ≠ (𝐵 + 𝐶) ↔ 𝐴 ≠ 𝐵)) |
7 | 1, 6 | mpbird 166 | 1 ⊢ (𝜑 → (𝐴 + 𝐶) ≠ (𝐵 + 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ≠ wne 2306 (class class class)co 5767 ℂcc 7611 + caddc 7616 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-resscn 7705 ax-1cn 7706 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-iota 5083 df-fv 5126 df-ov 5770 |
This theorem is referenced by: modsumfzodifsn 10162 |
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